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Question Number 186721 by Mingma last updated on 09/Feb/23
Commented by Mingma last updated on 09/Feb/23
How far is the arc from A to B? The side of the square is 2 in length
Answered by HeferH last updated on 09/Feb/23
Commented by HeferH last updated on 09/Feb/23
![1−a^2 = (2 + (√2) + a)(2−(√2)−a) 1−a^2 = [2 + ((√2) + a)][2 −((√2)+a)] 1−a^2 = 4 − ((√2) + a)^2 1 − a^2 = 4 −(2 + a^2 + 2a(√2)) 1 − a^2 = 2 − a^2 − 2a(√2) −1 = −2a(√2) a = (1/(2(√2))) = ((√2)/4) ⇒ 2 − (√2) − ((√2)/4) = 2 − ((5(√2))/4) = ((8 − 5(√2))/4)](Q186725.png)
$$\mathrm{1}−{a}^{\mathrm{2}} =\:\left(\mathrm{2}\:+\:\sqrt{\mathrm{2}}\:+\:{a}\right)\left(\mathrm{2}−\sqrt{\mathrm{2}}−{a}\right) \\ $$$$\:\mathrm{1}−{a}^{\mathrm{2}} \:=\:\left[\mathrm{2}\:+\:\left(\sqrt{\mathrm{2}}\:+\:{a}\right)\right]\left[\mathrm{2}\:−\left(\sqrt{\mathrm{2}}+{a}\right)\right] \\ $$$$\:\mathrm{1}−{a}^{\mathrm{2}} \:=\:\mathrm{4}\:−\:\left(\sqrt{\mathrm{2}}\:+\:{a}\right)^{\mathrm{2}} \\ $$$$\:\mathrm{1}\:−\:{a}^{\mathrm{2}} \:=\:\mathrm{4}\:−\left(\mathrm{2}\:+\:{a}^{\mathrm{2}} \:+\:\mathrm{2}{a}\sqrt{\mathrm{2}}\right) \\ $$$$\:\mathrm{1}\:−\:{a}^{\mathrm{2}} \:=\:\mathrm{2}\:−\:{a}^{\mathrm{2}} \:−\:\mathrm{2}{a}\sqrt{\mathrm{2}} \\ $$$$\:−\mathrm{1}\:=\:−\mathrm{2}{a}\sqrt{\mathrm{2}} \\ $$$$\:{a}\:=\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\:=\:\frac{\sqrt{\mathrm{2}}}{\mathrm{4}} \\ $$$$\:\Rightarrow\:\mathrm{2}\:−\:\sqrt{\mathrm{2}}\:−\:\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\:=\:\mathrm{2}\:−\:\frac{\mathrm{5}\sqrt{\mathrm{2}}}{\mathrm{4}}\:=\:\frac{\mathrm{8}\:−\:\mathrm{5}\sqrt{\mathrm{2}}}{\mathrm{4}}\: \\ $$